The $z$-component of the Néel vector is measurable by the anomalous Hall conductivity in altermagnets because time reversal symmetry is broken. On the other hand, it is a nontrivial problem how to measure the in-plane component of the Néel vector. We study the second-order nonlinear conductivity of a system made of the $d$-wave altermagnet with the Rashba interaction. It is shown that the quantum-metric induced nonlinear conductivity and the nonlinear Drude conductivity are proportional to the in-plane component of the Néel vector, and hence, the in-plane component of the Néel vector is measurable. We obtain analytic formulas of the quantum-metric induced nonlinear conductivity and the nonlinear Drude conductivity both for the longitudinal and transverse conductivities. The quantum-metric induced nonlinear conductivity diverges at the Dirac point, while the nonlinear Drude conductivity is always finite. Hence, the quantum-metric induced nonlinear conductivity is dominant at the Dirac point irrespective of the relaxation time.

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• Berry curvature
• Electrical conductivity
• Geometric & topological phases
• Rashba coupling
• Spin-orbit coupling
• Geometry
• Tight-binding model